图的ε-矩阵乘积分解
On $\varepsilon$-Matrix Product Factorization of graphs
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中文总结 AI 辅助
该研究针对图的精确矩阵乘积分解的刚性障碍，提出ε-近似分解方法，推导等价形式与定量约束，为完全图、树等图族构造了低误差的近似分解，证明精确障碍可在误差比例极小时消失。
中文摘要 AI 辅助
我们引入了图的矩阵乘积分解的近似版本。n个顶点的简单图G被称为容许ε-矩阵乘积分解，当且仅当存在同一顶点集上的简单图H和K，使得A(H)A(K)与A(G)在至多εn²个条目上不一致。这种汉明型松弛在误差集之外保留了每条边作为唯一H到K两步见证的精确解释。我们建立了等价的矩阵形式和见证形式，表明集合N_H(w)×N_K(w)构成了G的有序邻接关系的近似不交分解，并推导了涉及游走计数和因子图度数的定量约束。随后，我们为若干图族构造了近似分解：每个完全图K_n的矩阵乘积分解距离为O(1/n)，尽管精确同余障碍仅当n≡1 mod 4时容许精确分解；更一般地，固定r顶点图的膨胀图容许ε-分解且ε≤r/n，当每个非孤立部分为偶阶时构造是精确的；对于二分图，我们给出实现几乎所有边单向的单边分解，特别地，每个n≥2顶点的树容许ε-分解且ε≤1/n，尽管非平凡树均不可精确分解。这些结果表明，刚性的精确障碍可在条目误差比例消失时消失。
英文摘要
We introduce an approximate version of matrix product factorization for graphs. A simple graph $G$ on $n$ vertices is said to admit an $\varepsilon$-matrix product factorization if there exist simple graphs $H$ and $K$ on the same vertex set such that $A(H)A(K)$ and $A(G)$ disagree in at most $\varepsilon n^{2}$ entries. This Hamming-type relaxation preserves, outside the error set, the exact interpretation of each edge as having a unique $H$-then-$K$ two-step witness. We establish equivalent matrix, and witness formulations, showing that the sets $N_H(w)\times N_K(w)$ form an approximate disjoint decomposition of the ordered adjacency relation of $G$, and we derive quantitative constraints involving walk counts and the degrees of the factor graphs. We then construct approximate factorizations for several graph families. Every complete graph $K_n$ has matrix-product-factorization distance $O(1/n)$, despite the exact congruence obstruction that permits exact factorization only when $n\equiv 1\pmod 4$. More generally, a blow-up of a fixed graph on $r$ vertices admits an $\varepsilon$-factorization with $\varepsilon\le r/n$, and the construction is exact whenever every non-isolated part has even order. For bipartite graphs, we give one-sided factorizations that realize one orientation of almost all edges. In particular, every tree on $n\ge2$ vertices admits an $\varepsilon$-factorization with $\varepsilon\le 1/n$, although no nontrivial tree is exactly factorizable. These results show that rigid exact obstructions may disappear under a vanishing proportion of entrywise errors.
